Scattering theory for wave equations with long-range perturbations
type: Item
instance of: scholarly article
title: Scattering theory for wave equations with long-range perturbations
zbMATH Open document ID: 0586.35072
author: Hirokazu Iwashita
published in: Tsukuba Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: This work is devoted to the scattering theory for the wave equation with long-range perturbations: \[ Lu=\partial^ 2_ t u- \sum^{n}_{j,k=1}\partial_{x_ j} a^{jk}(x) \partial_{x_ k} u+V(x)u, \] where the symmetric matrix \(A(x)=(a^{jk}(x))\) is positively definite, V(x)\(\geq 0\) and \[ | \partial_ x^{\alpha}(a^{jk}(x)- \delta^{jk})| \leq C_{\alpha}(1+| x|)^{-| \alpha | -\delta},\quad 0<\delta <1;\quad | \partial_ x^{\alpha}V(x)| \leq C_{\alpha}(1+| x|)^{-| \alpha | -\delta} \] for all multi-indices \(\alpha\). The author proves the existence and completeness of the modified wave operators related to L and \(\partial^ 2_ t-\Delta\). For this purpose he establishes a general invariance principle involving as admissible functions real- valued \(C^{\infty}({\mathbb{R}}^+)\)-functions \(\phi\) (\(\lambda)\) with \(\phi '(\lambda)>0\) for all \(\lambda \in {\mathbb{R}}^+\).
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 35A07
Mathematics Subject Classification ID: 35Q99
zbMATH DE Number: 3939867
zbMATH Keywords: scattering theory
zbMATH Keywords: wave equation
zbMATH Keywords: wave operators
zbMATH Keywords: invariance principle
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496160194
OpenAlex ID: W615698524
DOI: 10.21099/TKBJM/1496160194
description: scientific article; zbMATH DE number 3939867